Yes, I just finished doing it about 5 minutes ago! I bought an AirPort Extreme and installed it as my WiFi network using the AirPort Utility on my MacBook Pro. Then I plugged in my Airport Express to a power socket upstairs at the other end of my house where I lose signal. The AirPort utility on my MacBook Pro recognized the Express and asked me if I wanted to use the Express to extend my existing network. I clicked yes...done! The whole thing took less than 10 minutes.

No you do not need both the airport express and a separate wireless router. The airport express is a wireless router by its self and doesn't need anything except the ethernet going in (from a modem or ethernet jack in the wall). Also I'm not sure what you tried to say at the very end but it is not 802.11ac yet (Hopefully an update in the near future!!!). The airport extreme is AC but the express is only N.

I have a lodge on a holiday park. The park itself has public wifi paid for by a subscription similar to hotels (when going online for the first time you are directed to a login page). The problem is, my lodge is on the edge of the park and I can only use the wifi in one corner of my living room, and not in the dining room or bedrooms. Could the AirPort Express login to the parks wifi and then have my MacBook Pro connect to the AirPort Express?

I have installed Mac OS X Server on my MacBook Air and would like to create a Wi-Fi network that asks for a username and password with the RADIUS server. Can I use the AirPort Express or do I have to buy the AirPort Extreme?